Fermiology in the underdoped high $T_{\rm c}$ cuprates presents us with unique challenges, requiring experimentalists to look deeper into the data than is normally required for clues. Recent measurements of an oscillatory chemical potential affecting the oscillations at high magnetic fields provide a strong indication of a single type of carrier pocket. When considered in conjunction with photoemission and specific heat measurements, a Fermi surface comprised almost entirely of nodal pockets is suggested. The mystery of the Fermi surface is deepened, however, by a near doping-independent Fermi surface cross-sectional area and negative Hall and Seebeck coefficients. We explore ways in which these findings can be reconciled, taking an important hint from the diverging effective mass yielded by quantum oscillations at low dopings. The author wishes to thank Suchitra Sebastian, Moaz Atarawneh, Doug Bonn, Walter Hardy, Ruixing Liang, Charles Mielke and Gilbert Lonzarich who have contributed to this work. The work is supported by the NSF through the NHMFL and by the DOE project ``Science at 100 tesla.'' [Preview Abstract]

The momentum-resolved nature of angle-resolved photoemission spectroscopy
(ARPES) has made it a key probe of emergent phases in the cuprates, such as
superconductivity and the pseudogap, which have anisotropic momentum-space
structure. ARPES can be used to infer the origin of spectral gaps from their
distinct phenomenology---temperature, doping, and momentum dependence, and
this principle has been used to argue that the pseudogap is a distinct phase
from superconductivity, rather than a precursor [1]. We have studied
Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\delta }$ (Bi-2212) using laser-ARPES, and
our data give evidence for three distinct quantum phases comprising the
superconducting ground state, accompanied by abrupt changes at p$\sim $0.076
and p$\sim $0.19 in the doping-and-temperature dependence of the gaps near
the bond-diagonal (nodal) direction [2]. The latter doping likely marks the
quantum critical point of the pseudogap, while the former represents a
distinct competing phase at the edge of the superconducting dome.
Additionally, we find that the pseudogap advances closer towards the node
when superconductivity is weak, just below T$_{c}$ or at low doping, and
retreats towards the antinode well below T$_{c}$ and at higher doping. This
phase competition picture together with the two critical doping are
synthesized into our proposed phase diagram, which also reconciles
conflicting phase diagrams commonly used in the field. Our results
underscore the importance of quantum critical phenomena to cuprate
superconductivity, provide a microscopic picture of phase competition in
momentum space, and predict the existence of phase boundaries inside the
superconducting dome which are different from simple extrapolations from
outside the dome.
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[1] I. M. Vishik, W. S. Lee, R.-H. He, M. Hashimoto, Z. Hussain, T. P.
Devereaux, and Z.-X. Shen. \textit{New J. Phys. }\textbf{12}, 105008 (2010).
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[2] I. M. Vishik, M. Hashimoto, R.-H. He, W. S. Lee, F. Schmitt, D. H. Lu,
R.G. Moore, C. Zhang, W. Meevasana, T. Sasagawa, S. Uchida, K. Fujita, S.
Ishida, M. Ishikado, Y. Yoshida, H. Eisaki, Z. Hussain, T. P. Devereaux, and
Z.-X. Shen, \textit{Submitted} (2011). [Preview Abstract]

The discovery of high T$_{c}$ superconductors has revived interest in
Anderson's resonating valence bond theory (RVB) of a quantum spin liquid,
first proposed in 1973. In the past few years, several examples of quantum
spin liquids have been discovered experimentally. The organic spin liquid
has been studied most thoroughly and shows strong evidence for emergent
fermionic spinons. I shall review some of the data and argue that theories
based on slave particles and gauge fields have been successful in accounting
for these remarkable data. The question remains as to whether a similar
formulation of fermionic spinon and bosonic holes can form the basis for a
theory of high T$_{c}$ superconductors. I shall show that a recent
modification\footnote{T. Senthil and P.A. Lee, Phys. Rev. Lett. \textbf{103}, 076402 (2009).} of the mean field RVB phase diagram can explain a lot of
the phenomenology. I shall also attempt to put this theory in the context of
recent discoveries concerning symmetry breaking in the pseudogap phase.
[Preview Abstract]